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GPW and GAPW Zurich, July 12th 2017 CP2K : GPW and GAPW Marcella Iannuzzi Department of Chemistry, University of Zurich http://www.cp2k.org Basis Set Representation KS matrix formulation when the wavefunction is expanded into a basis System size {Nel, M}, P [MxM], C [MxN] ⇥i(r)= Cαiφα(r) α Variational n(r)= fiCαiC⇥iφα(r)φ⇥(r)= P⇥φα(r)φ⇥(r) principle i ⇥ ⇥ Constrained minimisation P = PSP problem KS total energy E[ ψ ]=T [ ψ ]+Eext[n]+EH[n]+EXC[n]+EII { i} { i} Matrix formulation of the KS equations H xc K(C)C = T(C)+Vext(C)+E (C)+E (C)=SCε 2 Self-consistency SCF Method Generate a starting density ⇒ ninit Input 3D Coordinates init Generate the KS potential ⇒ VKS of atomic nuclei Solve the KS equations ⇒ , ψ � Initial Guess Molecular Orbitals Fock Matrix (1-electron vectors) Calculation 1 Calculate the new density ⇒ n 1 New KS potential ⇒ VKS Fock Matrix Diagonalization New orbitals and energies ⇒ �1 , ψ Calculate 2 Properties Yes SCF No New density ⇒ n Converged? End ….. until self-consistency 3to required precision Classes of Basis Sets Extended basis sets, PW : condensed matter Localised basis sets centred at atomic positions, GTO Idea of GPW: auxiliary basis set to represent the density Mixed (GTO+PW) to take best of two worlds, GPW Augmented basis set, GAPW: separated hard and soft density domains 4 GPW Ingredients linear scaling KS matrix computation for GTO Gaussian basis sets (many terms analytic) 2 mx my mz ↵mr ⇥i(r)= Cαiφα(r) φ↵(r)= dm↵gm(r) gm(r)=x y z e− α m X Pseudo potentials Plane waves auxiliary basis for Coulomb integrals Regular grids and FFT for the density Sparse matrices (KS and P) Efficient screening G. Lippert et al, Molecular Physics, 92, 477, 1997 J. VandeVondele et al, Comp. Phys. Comm.,167 (2), 103, 2005 5 Gaussian Basis Set Localised, atom-position dependent GTO basis ϕµ(r)= dmµgm(r) m CP2K: Ab initio Molecular Dynamics Simulations Towards Linear Scaling HF/Exact Exchange Summary Acknowledgment Expansion of the density using the density matrix Sparse Matrices n(r)= Pµν ϕµ(r)ϕν∗(r) µν Gaussian basis: • Operator matricesOperator are rather matrices sparse are sparse The sparsity of H and S Sαβ=∫ϕα(r)ϕβ (r)dr Sµ⌫ = 'µ(r)'⌫ (r)dr H αβ=∫ ϕα (r)v(r)ϕβ (r)dr r ϕ (r) Z ϕα ( ) β The sparsity pattern of S and H depends on the basis and the spatial location of the atoms, but not on theH µchemical⌫ = properties'µ(r)V ( ofr) the'⌫ (r)dr system in GGAZ DFT. Sαβ HIV-1 Protease-DMP323 complex in solution (3200 atoms) 6 • Orbital matrices are invariant under unitary transformation The overlap (integral of the product) rapidly decays with the spatial separation of the basis Chemical localization: Boys, Edminston-Rudenberg, etc. functions. Mathematical localization Analytic Integrals Cartesian Gaussian n n n η(r R)2 g(r, n, η, R)=(x R ) x (y R ) y (z R ) z e− − − x − y − z Differential relations @ @ @ n)=2⌘ n + 1i) ni n 1i) n)= n) @Ri | | − | − @Ri | −@ri | Obara-Saika recursion relations (0 (r) 0 ) (a + 1i (r) b) a|O | b |O | Obara and Saika JCP 84 (1986), 3963 7 Basis Set library GTH_BASIS_SETS ; BASIS_MOLOPT ; EMSL_BASIS_SETS O 6-31Gx 6-31G* 4O SZV-MOLOPT-GTHO SZV-GTH SZV-MOLOPT-GTH-q6 1 0 10 6 1 2 0 1 7 1 1 5484.67170000 2 0 1 4 1 1 0.00183110 O 6-311++G3df3pd 6-311++G(3df,3pd) 12.015954705512 -0.060190841200 0.036543638800 825.23495000 8.3043855492 0.01395010 0.1510165999 -0.0995679273 9 5.108150287385 -0.129597923300 0.120927648700 188.04696000 2.4579484191 0.06844510 -0.0393195364 -0.3011422449 1 0 0 6 1 2.048398039874 0.118175889400 0.251093670300 52.96450000 0.23271430 8588.50000000 0.00189515 0.832381575582 0.7597373434 0.462964485000 -0.6971724029 0.352639910300 -0.4750857083 16.89757000 0.47019300 1297.23000000 0.01438590 0.352316246455 0.2136388632 0.450353782600 -0.3841133622 0.294708645200 -0.3798777957 5.79963530 0.35852090 299.29600000 0.07073200 #0.142977330880 0.092715833600 0.173039869300 1 0 1 3 1 1 87.37710000 0.24000100 O0.046760918300 DZVP-GTH -0.000255945800 0.009726110600 15.53961600 -0.11077750 0.07087430 25.67890000 0.59479700 # 23.59993360 -0.14802630 0.33975280 3.74004000 0.28080200 O DZVP-MOLOPT-GTH DZVP-MOLOPT-GTH-q6 21.01376180 0 1 4 2 2 1.13076700 0.72715860 1 0 1 3 1 1 1 1 0 1 1 8.3043855492 1 1 0.1510165999 0.0000000000 -0.0995679273 0.0000000000 42.11750000 0.11388900 0.03651140 2 0 2 7 2 2 1 0.27000580 2.4579484191 1.00000000 -0.0393195364 1.00000000 0.0000000000 -0.3011422449 0.0000000000 9.62837000 0.92081100 0.23715300 12.015954705512 -0.060190841200 0.065738617900 0.036543638800 -0.034210557400 0.014807054400 1 2 2 1 0.7597373434 1 -0.6971724029 0.0000000000 -0.4750857083 0.0000000000 2.85332000 -0.00327447 0.81970200 5.108150287385 -0.129597923300 0.110885902200 0.120927648700 -0.120619770900 0.068186159300 0.80000000 1.00000000 1 0 1 1 1 1 2.048398039874 0.2136388632 0.118175889400 -0.3841133622 -0.053732406400 1.0000000000 0.251093670300 -0.3798777957 -0.213719464600 1.0000000000 0.290576499200 # 0.90566100 1.00000000 1.00000000 0.832381575582 3 2 2 1 1 0.462964485000 -0.572670666200 0.352639910300 -0.473674858400 1.063344189500 O 6-31Gxx 6-31G** 1 0 1 1 1 1 0.352316246455 1.1850000000 0.450353782600 1.0000000000 0.186760006700 0.294708645200 0.484848376400 0.307656114200 4 0.25561100 1.00000000 1.00000000 #0.142977330880 0.092715833600 0.387201458600 0.173039869300 0.717465919700 0.318346834400 1 0 0 6 1 1 2 2 1 1 0.046760918300 -0.000255945800 0.003825849600 0.009726110600 0.032498979400 -0.005771736600 O5484.67170000 TZVP-GTH 0.00183110 5.16000000 1.00000000 # 825.23495000 2 0.01395010 1 2 2 1 1 O TZVP-MOLOPT-GTH TZVP-MOLOPT-GTH-q6 188.04696000 2 0 1 5 3 3 0.06844510 1.29200000 1.00000000 1 52.96450000 10.2674419938 0.23271430 0.0989598460 0.0000000000 0.0000000000 -0.0595856940 1 2 2 1 10.0000000000 0.0000000000 2 0 2 7 3 3 1 16.89757000 3.7480495696 0.47019300 0.1041178339 0.0000000000 0.0000000000 -0.1875649045 0.32250000 0.0000000000 1.00000000 0.0000000000 12.015954705512 -0.060190841200 0.065738617900 0.041006765400 0.036543638800 -0.034210557400 -0.000592640200 0.014807054400 5.79963530 0.35852090 1 3 3 1 1 5.108150287385 1.3308337704 -0.129597923300 -0.3808255700 0.110885902200 0.0000000000 0.080644802300 0.0000000000 0.120927648700 -0.3700707718 -0.120619770900 0.0000000000 0.009852349400 0.0000000000 0.068186159300 1 0 1 3 1 1 1.40000000 1.00000000 2.048398039874 0.4556802254 0.118175889400 -0.6232449802 -0.053732406400 1.0000000000 -0.067639801700 0.0000000000 0.251093670300 -0.4204922615 -0.213719464600 1.0000000000 0.001286509800 0.0000000000 0.290576499200 15.53961600 -0.11077750 0.07087430 1 0 1 1 1 1 0.832381575582 0.1462920596 0.462964485000 -0.1677863491 -0.572670666200 0.0000000000 -0.435078312800 1.0000000000 0.352639910300 -0.2313901687 -0.473674858400 0.0000000000 -0.021872639500 1.0000000000 1.063344189500 3.59993360 -0.14802630 0.33975280 0.08450000 1.00000000 1.00000000 0.352316246455 31.01376180 2 2 1 1 0.450353782600 1.13076700 0.186760006700 0.72715860 0.722792798300 0.294708645200 0.484848376400 0.530504764700 0.307656114200 1 0 0.142977330880 1 1 1.1850000000 1 1 0.092715833600 1.0000000000 0.387201458600 -0.521378340700 0.173039869300 0.717465919700 -0.436184043700 0.318346834400 0.046760918300 0.27000580 -0.000255945800 1.00000000 0.003825849600 1.00000000 0.175643142900 0.009726110600 0.032498979400 0.073329259500 -0.005771736600 1 2 2 1 1 0.80000000 1.00000000 8 GTO in CP2K The repository contains several GTO libraries cp2k/data/ ALL_BASIS_SETS BASIS_RI_cc-TZ GTH_POTENTIALS dftd3.dat ALL_POTENTIALS BASIS_SET HFX_BASIS nm12_parameters.xml BASIS_ADMM BASIS_ZIJLSTRA HF_POTENTIALS rVV10_kernel_table.dat BASIS_ADMM_MOLOPT DFTB MM_POTENTIAL t_c_g.dat BASIS_LRIGPW_AUXMOLOPT ECP_POTENTIALS NLCC_POTENTIALS t_sh_p_s_c.dat BASIS_MOLOPT EMSL_BASIS_SETS POTENTIAL vdW_kernel_table.dat BASIS_MOLOPT_UCL GTH_BASIS_SETS README Tools for the optimisation of GTO basis sets are available in cp2k, based on atomic and molecular electronic structure calculations 9 Pseudopotentials Outline Recap of Previous lecture All electrons vs pseudopotentials The Hartree-Fock-Kohn-Sham method Core electronsClasses of Basis-set are eliminated ZV=Z-Zcore The exchange and correlation hole Condensed phase: Bloch’s th and PBC Solving the electronic problem in practice Atomic 1s : exp{-Z r} Smooth nodeless pseudo-wfn close to nuclei Bare Coulomb replaced by screened Coulomb Inclusion of relativistic effects Transferable Angular dependent potentials: Pt p peaked at 3.9Å s peaked at 2.4Å 10 d peaked at 1.3Å Marialore Sulpizi Density Functional Theory: from theory to Applications GTH Pseudopotentials Norm-conserving, separable, dual-space Local PP : short-range and long-range terms 4 (2i 2) 2 PP PP PP − αPPr Zion PP V (r)= C (2)α r e−( ) erf α r loc i − r i=1 ⇧ ⇤⌃analytically⌅ part of ES ⇥ Non-Local PP with Gaussian type projectors Accurate and PP lm l lm V (r, r0)= r p h p r0 nl h | i i ijh j | i Transferable ij Xlm X Scalar 2 1 r lm l lm (l+2i 2) relativistic r p = N Y (ˆr) r − e− 2 rl | i i “ ” ⇥ Few parameters Goedeker, Teter, Hutter, PRB 54 (1996), 1703; Hartwigsen, Goedeker, Hutter, PRB 58 (1998) 3641 11 Electrostatic Energy Periodic system PP n˜⇤(G)˜n(G) 1 ZAZB EES = Vloc (r)n(r)dr +2⇡⌦ 2 + G 2 RA RB G A=B Z X X6 | − | total charge distribution ntot(r)=n(r)+ nA(r) including n(r) and Z A r R − A ZA rc A ZA r RA n (r)= π 3/2 e A V (r)= erf | − | A c 3 − „ « core − r R rc −(rA) | − A| A ⇥ c PP rA = √2 rlocA cancels the long range term of local PP H E [ntot] long range SR 1 ntot(r)ntot(r) EES = Vloc (r)n(r)+ drdr 2 r r smooth ⌅ ⌅⌅ | − | 1 Z Z R R 1 Z2 + A B erfc | A − B A c c c 2 RA RB 2 2 − ⇥ r A=B (rA) +(rB) ⇥ A 2π A ⇤⇥ | − | ⇤ ⇧ Eov short range, pair Eself 12 Auxiliary Basis Set Long range term : Non-local Hartree potential H 1 ntot(r)ntot(r) E [n ]= drdr tot 2 r r | − | Orthogonal, unbiased, naturally periodic PW basis Efficient Mapping 1 iG r n˜(r)= n˜(G) e · FFT Ω G Grid spacing [Å] 0.15 0.13 0.11 0.10 0.09 0.08 -1 10 H2O, GTH, TZV2P -2 10 -3 Electrostatic Linear scaling solution of the Poisson10 equation -4 10 Energy n˜ (G)˜n -5 (G) H tot∗ Error [a.u.] 10tot E [ntot]=2πΩ -6 G210 G -7 10 100 200 300 400 500 Plane wave cutoff [Ry] 13 Fig.
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